A manufacturing process for a widget has four stages and involves $4$ parts. In Section A parts $1$ and $2$ are assembled. In Section B parts $3$ and $4$ are assembled. In Section C the two assemblies are welded together and finally in Section D the widget is painted and packaged. Sections A and B have $5$ assembly machines each. Each assembly machine in Section A has an output of $50$ parts per hour and each machine in section B has an output of $60$ parts per hour. Section C has $8$ welding machines each of which has an output of $40$ parts per hour and Section D has $6$ painting machines each of which has an output of $75$ parts per hour. All machines need one worker each to operate. Purchase costs of the parts are follows:

Cost of purchasing part $1$ $$0\mathrm{.}02$

Cost of purchasing part $2$ $$0\mathrm{.}05$

Cost of purchasing part $3$ $$0\mathrm{.}01$

Cost of purchasing part $4$ $$0\mathrm{.}09$

Cost of assembly in section A is $0\mathrm{.}30/$part$,$ cost of assembly in section B is $0\mathrm{.}40/$part$,$ cost of welding is $0\mathrm{.}35/$part and cost of painting is $0\mathrm{.}10/$part$.$

At present, the manufacturing process runs for $100$ hours a month excluding scheduled maintenance times.

a$)$What is the output of this process? What is the bottleneck? What is the total cost of the process? What is the cost per unit?

b$)$If we can add another shift to one section in the process to increase its capacity, where would it be$?$ What is the output of this process? What is the new bottleneck? What is the total cost of the process? What is the cost per unit?

c$)$If on top of doubling the shift as discussed in part $($b$),$ if we add four extra assembly machines to Section B and $3$ extra welding machines to section C what is the output of this process? What is the bottleneck? What is the total cost of the process? What is the cost per unit?