In the semiconductor and liquid crystal display panel $($LCD$)$ industries, automated guided vehicles $($AGVs$)$ are widely used to transfer a lot $($a cassette full of wafers or panes of glass$)$ between processing machines. Figure $1$ shows AGVs operating in a fab producing LCD panels $($source: Shinsung ENG$).$

Due to their complicated mechanisms and high utilization, these AGVs are subject to failures. When an AGV fails, an AGV service crew in the factory takes the broken AGV to the repair shop and fixes it$.$ The crew is stationed at a single location near the factory. AGV breakdowns occur randomly $($according to a Poisson process$)$ at a mean rate of one per hour. The time required to fix an AGV has a negative exponential distribution $($for any crew size$).$ The expected repair time required by a one$-$worker crew is $2$ hours. The cost per hour for each member of a repair crew is $$10\mathrm{.}00.$ The cost that is attributable to not having an AGV in use $($e$.$g$.,$ an AGV in the repair shop$)$ is estimated to be $$40\mathrm{.}00$ per hour. This cost is due to a production loss caused by a lack of AGVs.

Assume that the mean service rate of the repair crew is proportional to its size. What should the crew size be in order to minimize the expected total cost of this operation per hour? $($Hint: You will need to solve for the proportionality constant$)($Round your answer to a full number$)$

Assume that the mean service rate is proportional to the square root of the crew size. What should the crew size be in order to minimize the expected total cost of this operation per hour? $($Hint: You will need to solve for the proportionality constant$)($Round your answer to a full number$)$