$\backslash$table$[[$ACTIVITY$,$ACTIVITY DESCRIPTION,$\backslash$table$[[$IMMEDIATE$],[$PREDECESSORS$]],$TIME$],[$A$,$Attach wheels to tub,$-,5],[$B$,$Attach motor to lid,$-,1\mathrm{.}5],[$C$,$Attach battery pack,B$,3],[$D$,$Attach safety cutoff,C$,4],[$E$,$Attach filters,B$,3],[$F$,$Attach lid to tub,A$,$ E$,2],[$G$,$Assemble attachments,$-,3],[$H$,$Function test,D$,$ F$,$ G$,3\mathrm{.}5],[$I$,$Final inspection,H$,2],[$J$,$Packing,I,$2]]$

The following table details the tasks required for Indiana$-$based Frank Pianki Industries to manufacture a fully portable industrial vacuum cleaner. The times in the table are in minutes. Demand forecasts indicate a need to operate with a cycle time of $10$ minutes.

a$)$ Draw the appropriate precedence diagram for this production line.

b$)$ Assign tasks to workstations and determine how much idle time is present each cycle.

c$)$ Discuss how this balance could be improved to $100\%.$

d$)$ What is the theoretical minimum number of workstations?