Module $3$ Application Activity $$ Heetjet $$ Chapter $7$

I. The specification for the distance between nozzles on the burner of a high$-$efficiency furnace produced by Heetjet Enterprises is $15\mathrm{.}00\backslash$pm $0\mathrm{.}025$ mm$.$ The loss due to a defective burner unit is $$40.$ A sample of $30$ burners was drawn from the production process and the results, in millimeters, can be found below.

a$.$ Compute the mean and standard deviation from the sample data below and use it to compute the value of $$k$$ in the Taguchi loss function.

b$.$ Using the formula in the text, what is the expected loss per part from this process?

$15\mathrm{.}00815\mathrm{.}00015\mathrm{.}00814\mathrm{.}98514\mathrm{.}99315\mathrm{.}008$

$15\mathrm{.}00015\mathrm{.}00015\mathrm{.}01514\mathrm{.}99315\mathrm{.}00014\mathrm{.}993$

$14\mathrm{.}98514\mathrm{.}99315\mathrm{.}00815\mathrm{.}00015\mathrm{.}00015\mathrm{.}015$

$15\mathrm{.}00014\mathrm{.}99315\mathrm{.}00815\mathrm{.}00815\mathrm{.}00015\mathrm{.}000$

$15\mathrm{.}00015\mathrm{.}01515\mathrm{.}00015\mathrm{.}00015\mathrm{.}00814\mathrm{.}985$

II$.$ The furnaces have a failure rate of $=0\mathrm{.}00005.$ Find the mean time to failure $($MTTF$).$ What is the probability $($assuming an exponential probability distribution$)$ that the component will not have failed after $7,500$ hours of operation?

III. The manufacturing process for the furnaces has three operations that are performed in series. Because of the nature of the process, machines frequently fall out of adjustment and must be repaired. To keep the system going, two identical machines are used at each stage; thus, if one fails, the other can be used while the first is repaired $($see accompanying figure$).$

The reliabilities of the machines are as follows:

Machine Reliability

A $0\mathrm{.}91$

B $0\mathrm{.}87$

C $0\mathrm{.}94$

a$.$ Analyze the system reliability, assuming only one machine at each stage $($all the backup machines are out of operation$).$

b$.$ How much is the reliability improved by having two machines at each stage?