This problem related to Manufacturing Systems subject.

Production is set to occur on a two$-$machine line with deterministic processing times and geometric

repair and failure times, with operation$-$dependent failures. The operation time is $1$ minute per

machine. The mean time to failure of each machine is $16$ hours and $40$ minutes.$|$The mean down time

is $1$ hour, $51$ minutes, and $7$ seconds.

Part $1$

Calculate the production rate of the following lines in parts per minute.

a$)$ What is the production rate of a zero$-$buffer line?

b$)$ What is the production rate of an infinite buffer line?

Answer in $4$ decimal places.

Part $2$

Assume the factory works $50$ weeks, with $40$ hours per week on average.

a$)$ What is the smallest buffer that allows an expected production rate of $106,000$ items$/$year$.$

Round to the nearest integer.

Part $3$

The factory gets $$1$ for each item sold, and it costs $$20$ to hold an item in inventory.

a$)$ What is the profit if the factory sizes the buffer to make $106,000$ units?

b$)$ What is the optimal buffer size for maximizing profit?

c$)$ What is the optimal profit?

Round to the nearest integer.