$($Please do on excel$)$ Kirkpatrick Aircrafts operates a large number of computerized plotting machines. For the most part, the plotting devices are used to create line draw$-$ ings of complex wing airfoils and fuselage part di$-$ mensions. The engineers operating the automated plotters are called loft lines engineers.

The computerized plotters consist of a mini$-$ computer system connected to a $4-$ by $5-$foot flat table with a series of ink pens suspended above it$.$ When a sheet of clear plastic or paper is properly placed on the table, the computer directs a series of horizontal and vertical pen movements until the desired figure is drawn.

The plotting machines are highly reliable, with the exception of the four sophisticated ink pens that are built in$.$ The pens constantly clog and jam in a raised or lowered position. When this occurs, the plotter is unusable.

Currently, Kirkpatrick Aircrafts replaces each pen as it fails. The service manager has, however, proposed replacing all four pens every time one fails. This should cut down the frequency of plotter failures. At present, it takes one hour to replace one pen. All four pens could be replaced in two hours. The total cost of a plotter being unusable is $$500$ per hour. Each pen costs $$80.$ The following break$-$ down data are thought to be valid: a$.$For each option $($replacing one pen at a time and replacing all four pens at a time$),$ simulate the average total time a plotter would operate before it would have $20$ failures. Then compute the total cost per hour for each option to determine which option Kirkpatrick Aircrafts should use. Use N replications.

$($b$)$ Compute the total cost per hour analytically for each option. How do these results compare to the simulation results?