Positively Rivet Inc. is a small machine shop that produces sheet metal products. It had one line dedicated to the manufacture of light$-$duty vent hood shells, but because of strong demand it recently added a second line. The new line makes use of higher$-$capacity automated equipment but consists of the same basic four processes as the old line. In addition, the new line makes use of one machine per workstation, while the old line has parallel machines at the workstations. The processes, along with their machine rates, number of machines per station, and average times for a lone job to go through a station $($i$.$e$.,$ not including queue time$),$ are given for each line in the following table:$$Process

Old Line

New Line

Rate per Machine $($parts$/$hour$)$

Number Machines per Station

Time $($minute$)$

Rate per Machine $($parts$/$hour$)$

Number Machines per Station

Time $($minute$)$

Punching Braking Assembly Finishing

$15122050$

$4421$

$4\mathrm{.}05\mathrm{.}03\mathrm{.}01\mathrm{.}2$

$120$

$120$

$125$

$125$

$1111$

$0\mathrm{.}500\mathrm{.}500\mathrm{.}480\mathrm{.}48$Over the past $3$ months, the old line has averaged $315$ parts per day, where one day consists of one $8-$hour shift, and has had an average WIP level of $400$ parts. The new line has averaged $680$ parts per $8-$hour day with an average WIP level of $350$ parts. Management has been dissatisfied with the performance of the old line because it is achieving lower throughput with higher WIP than the new line. Your job is to evaluate these two lines to the extent possible with the above data and identify potentially attractive improvement paths for each line by addressing the following questions.

$($a$)$ Compute rb$,$ T$0,$ and W$0$ for both lines. Which line has the larger critical WIP? Explain why.

$($b$)$ Compare the performance of the two lines to the practical worst case. What can you conclude about the relative performance of the two lines compared to their underlying capabilities? Is management correct in criticizing the old line for inefficiency?

$($c$)$ Ifyouwerethemanagerinchargeoftheselines,whatoptionwouldyouconsiderfirstto improve throughput of the old line? Of the new line?