BridgeRock is a major manufacturer of tires in the U.S.. The
company had five manufacturing facilities where tires were made and
another 20 facilities for various components and materials used in
tires. Each manufacturing facility produced 10,000 tires every
hour. Quality had always been emphasized at BridgeRock, but lately
quality was a bigger issue because of recent fatal accidents
involving tires made by other manufacturers due to tread
separation. All tire manufacturers were under pressure to ensure
problems did not arise in the future. However, even with multiple
visual quality inspections in place, adhesion flaws and other
internal problems were not visible, so manufacturers randomly
pulled tires from the production line and cut them apart to look
for defects. Given the large number of steps in building a tire,
errors tended to accumulate, which possibly results in larger
process quality variability. Another issue was related to the
settings on various machines. Over time, these settings tended to
vary because of wear and tear on the machines. In such a situation,
a machine would produce defective product even if the machine had
the correct setting.
To detect process variations, the company implemented a
statistical process control (SPC) program. At the extruder, the
rubber for the AX-123 tires had thickness specifications of 400±10
thousandth of an inch (thou) (these are the specification limits).
The quality team at BridgeRock analyzed many samples of output from
the extruder and determined that if the extruder settings were
accurate, the output produced by the machine had a thickness that
was normally distributed with a mean of 400 thou and a standard
deviation of 4 thou.
Answer the following questions:
a. If the extruder setting is accurate, what proportion of the
rubber extruder will be with specifications?
b. The quality team asked operators to take a sample of 10
sheets of rubber each hour from the extruder and measure the
thickness of each sheet. Based on the average thickness of this
sample, operators will decide if the extrusion process is in
control or not. Given that z=3 for constructing control limits,
what upper and lower control limits should they specify to the
operators?
c. If a bearing is worn out, the extruder produces a mean
thickness of 403 thou when the setting is 400 thou. The standard
deviation is still 4 thou. Under this condition, what proportion of
defective sheets will the extruder produce? Assuming the control
limits in question b, what is the probability that a sample taken
from the extruder with the worn bearings will be out of control? On
average, how many hours are likely to go by before the worn bearing
is detected?
d. Now consider the case where extrusion is a six sigma
process. In this case, the extruder output should have a mean of
400 thou and a standard deviation of 1.667 thou. Still using the
same 400±10 thou quality specifications, what proportion of the
rubber extruded will be within specifications in this case?
e. Assuming the operators will continue to collect samples of
10 sheets each hour to check if the process is in control, what
control limits should they set for the case when extrusion is a six
sigma process (keep in mind Z=3)?
f. Return to the case of the worn bearings in question c where
extrusion produces a mean thickness of 403 thou when the setting is
400 thou. Now the process standard deviation is 1.667 thou. Under
this condition, what proportion of defective sheets will the
extruder produce? Assuming the control limits calculated in
question e, what is the probability that a sample taken from the
extruder with the worn bearings will be out of control? On average,
how many hours are likely to go by before the worn bearing is
detected?
Mini-Case Hint Information
Before analyzing this case, you need to review the key
concepts we discussed in the class. Please pay attention to the
following items:
a. USL and LSL are relatively fixed and requested by internal
or external customers (We use these for defining defects). At the
same time, UCL and LCL are control limits which are usually
prepared by quality management professionals (for judging process
status). These two sets are quite different (and irrelevant).
b. The sigma quality (say, six sigma quality with 3.4 DPMO)
essentially refers to the magnitude of process standard deviation
(or variance). On the other hand, in the UCL and LCL formulas, the
number of sigma (Z value) relates to the corresponding confidence
level or type I error. For example, in most cases, 3 sigma is used,
which shows the type I error is about 0.26% or 0.3% (it means if
you spot one dot is located above UCL or below LCL, 0.3% of the
chance your out of control conclusion is wrong). Z value can be
chosen based on your required confidence level (in reality, Z=3 is
always used for constructing SPC charts everywhere, including in
this class), regardless of the sigma quality of the process (say 4
or 5 or 6 sigma) you want to control. You can see here that the
same z value means two different things in SPC analysis.
c. The process (or population) standard deviation is different
from the sample standard deviation. The sample standard deviation
is usually smaller. (Refer to the formula for making the
conversion: ). The fundamental difference is the sample standard
deviation measures the sample behavior (including n units), and the
process standard deviation measures individual product behavior
(single unit).
d. When judging if the process produces defects, you are
looking at one unit randomly picked product, to see if the quality
dimension is above USL or below LSL. The standard deviation is the
overall process (population) standard deviation.
e. When deciding if the process is out of control of not, you
are looking at one randomly picked sample (containing n=10 units of
product in the case), and judging if the sample mean is above UCL
or below LCL. UCL and LCL are computed based on X-double bar ± 3*
sample standard deviation. The sample standard deviation is
calculated in part c (see the conversion formula).
Please use Excel functions provided below for making the
calculations in Excel (It is easier and more accurate). You need to
present your answers for all the six questions raised in the case
here. If you just provide final answers for these questions, please
make sure that you explain what these numbers are, and also attach
an appendix for Excel worksheet to show your work.
Hereafter I provide hint information for each of the six case
questions:
1. Since the product specifications are given, we can easily
derive the USL and LSL values for the rubber sheets extruded. Based
on the distribution of the extruder output (mean and sigma are
given at the end of the case), we can find the probability of
non-defective products by using either Standard Normal Distribution
Table or Excel (for accuracy purpose, Excel is preferred, see the
following formula).
Proportion of output within specifications
= NORMDIST (USL, mean, sigma, 1) – NORMDIST (LSL, mean, sigma,
1)
Please note that Excel function NORMDIST(X, mean, sigma, 1) is
corresponding to the following shaded area:
2. In this case, the sample size is 10 sheets. As we know the
process standard deviation is 4 thou, we can easily compute the
sample standard deviation (Using the formula on page 1).
Consequently, the control charts can be constructed (using the
general X-bar chart formulas with known process SD on our lecture
PPT slide).
3. If a bearing is worn out, the output distribution becomes a
normal distribution with a mean of 403 thou and a standard
deviation (sigma) of 4 thou. Use the formula in (1) for estimating
the proportion of non-defective sheets (remember to update the mean
and sigma values). And then calculate the defective rate.
Now, we want to assess the chance a sample taken from the
extruder will be out of control, we need to focus on the sample
mean (403 thou) and the sample standard deviation (you calculated
in step (2)). The probability can be identified by using the
following formula:
Proportion of output within specifications
= NORMDIST (UCL, sample mean, sample sigma, 1) – NORMDIST
(LCL, sample mean, sample sigma, 1)
If the probability that a sample taken will be within control
limits is X, we know the probability of having an out-of-control
sample is (1-X). On average, it will take 1/(1-X) samples before an
out-of-control sample appears. This is the number of hours which
are likely to go by before the worn bearing is detected assuming
one sample will be taken per hour.
4. With the six sigma quality, the process output is normally
distributed with a mean of 400 and a standard deviation of 1.667.
Use the formula in (1) for estimating the probability. (Please
choose at least 8 decimal places for the relevant cells in Excel,
otherwise you are going to see 100% instead of 99.999999%)
5. Use the assumption that process output standard deviation
is 1.667 thou. Then repeat (2) steps. Calculate the new sample
sigma. What Z value do you want to use for constructing the control
limits? (The answer is already provided in this teaching note) Does
Z=6 make sense? Why? What is going to happen if you choose
Z=6?
6. Similar to (3), repeat all steps with different mean and
sigma. Compare the results and discuss the power of six sigma
quality. The formulas are all the same, but the standard deviations
and control limits are different.
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