Case: Quality Management $-$ Toyota $-$ Quality Control Analytics

As part of the process for improving the quality of their cars, Toyota engineers have identified a potential improvement to the process

that makes a washer that is used in the accelerator assembly. The tolerances on the thickness of the washer are fairly large since the

fit can be loose, but if it does happen to get too large, it can cause the accelerator to bind and create a potential problem for the

driver. Let's assume that, as a first step to improving the process, a sample of $40$ washers coming from the machine that produces the

washers was taken and the thickness measured in millimeters. The following table has the measurements from the sample:

Sample $12345678910$

$11\mathrm{.}921\mathrm{.}91\mathrm{.}82\mathrm{.}21\mathrm{.}721\mathrm{.}91\mathrm{.}71\mathrm{.}8$

$21\mathrm{.}82\mathrm{.}22\mathrm{.}12\mathrm{.}21\mathrm{.}91\mathrm{.}82\mathrm{.}11\mathrm{.}61\mathrm{.}81\mathrm{.}6$

$32\mathrm{.}12\mathrm{.}42\mathrm{.}22\mathrm{.}12\mathrm{.}121\mathrm{.}81\mathrm{.}71\mathrm{.}91\mathrm{.}9$

$42\mathrm{.}122\mathrm{.}41\mathrm{.}72\mathrm{.}221\mathrm{.}622\mathrm{.}12\mathrm{.}2$

$1.$ If the specification is such that no washer should be greater than $2\mathrm{.}4$ millimeters, assuming that the thicknesses are distributed normally, what fraction of the output is expected to be greater than this thickness? Round your answer to $6$ decimals.

$2.$ What if there is an upper and lower specification, where the upper thickness limit is $2\mathrm{.}4$ and the lower thickness limit is $1\mathrm{.}4?$ What fraction of the output is expected to be out of tolerance? Round your answer to six decimals.

$3.$ What is the Cpk for the process? Round your answer to $4$ decimals.

$4.$ What would be the Cp$$for the process if it were centered $($assume the process standard deviation is the same$)?$ Round your answer to $3$ decimals.$$

$5.$ What$$percentage$$of output would be expected to be out of tolerance if the process were centered? Round your answer to $4$ decimals.

$6.$ Setup X$-$bar and Range control charts for the current process.$$ Assume the operators will take samples of $10$ washers at a time.

What is The upper control limit for the X$-$bar chart? Round your answer to $3$ decimals.

$7.$ Plot the data on your control charts.$$ Does the current process appear to be in control?

a$.$ No$,$ the process is clearly out of control, with multiple data points beyond control limits$.$

b$.$ Yes, the process appears to be in control with no trends or concerns.

c$.$ No$,$ the process shows signs of instability, with significant variability but no observable trends.

d$.$ Yes, the process appears to be in control, but it$$s challenging to conclude definitively due to the limited number of samples. Additionally, a trend in both process mean and variability may be emerging.

$8.$ If the process could be improved so that the standard deviation was only about $\mathrm{.}10$ millimeters, what would be the best that could be expected with the processes relative to fraction defective? What is the value of DPMO? Round your answer to $3$ decimals.