Consider a production line with three single$-$machine stations in series. Each has processing times with mean $2$ hours and standard deviation of $2$ hours. $($Note that this makes it identical to the line represented in the practical worst case of Chapter $7.)$ Suppose the marginal profit is $$50$ per piece and the cost of WIP is $$0\mathrm{.}25$ per piece per hour.

$($a$)$ Compute the optimal throughput level operating as a push system and the optimal WIP level operating as a CONWIP system. What is the difference in the resulting profit levels? $(30$ Points$)$

$($b$)$ Suppose the process times actually have a mean and standard deviation of $2\mathrm{.}2$ hours, but the throughput used for the push system and the WIP level used for the pull system are computed as if the process times had a mean and standard deviation of $2$ hours $[$i$.$e$.,$ were equal to the levels computed in $($a$)].$ Now what is the profit level in the push and pull systems, and how do they compare? Repeat this calculation for a system in which processing times have a mean and standard deviation of $2\mathrm{.}4$ hours. What happens to the gap between the profit in the push and pull systems?