Que $1$: Quality Control in Aircraft Component Manufacturing $($See Notes $5)$

In an aerospace manufacturing plant, two processes, Process X and Process Y $,$ are used to produce a critical engine component. The required specification for the diameter of this component is $2\mathrm{.}00+-0\mathrm{.}01$ inches. Any deviation from the target diameter of $2\mathrm{.}00$ inches could affect the component's performance in high$-$stress conditions, which is critical for safety in aerospace applications.

The diameter produced by Process X is not uniform and is represented by the following probabilities:

$\backslash$table$[[$Diameter $($inches$),1\mathrm{.}99,2\mathrm{.}00,2\mathrm{.}01],[$Probability$,0\mathrm{.}25,0\mathrm{.}50,0\mathrm{.}25]]$

The diameter distribution of Process Y is represented by the following probabilities:

$\backslash$table$[[$Diameter $($inches$),1\mathrm{.}98,1\mathrm{.}99,2\mathrm{.}00,2\mathrm{.}01,2\mathrm{.}02],[$Probability$,0\mathrm{.}10,0\mathrm{.}20,0\mathrm{.}40,0\mathrm{.}20,0\mathrm{.}10]]$

The quality department at the aerospace plant uses the Taguchi loss function to calculate the potential cost associated with deviations from the target diameter, assuming $k=$$$200,000.$

a$)$ Calculate the expected loss for each process using the given probability distributions and loss function.

b$)$ Based on the expected loss, recommend which process $($ X or Y $)$ should be preferred to minimize quality$-$related costs.