Consider a paper mill with two paper machines. There are $5$ different types of paper that have to be produced. Items $1$ and $2$ have to be produced on machine $1$ and item $3$ has to be produced on machine $2.$ Items $4$ and $5$ can be produced on either one of the two machines.

$($a$)$ Determine the optimal rotation schedule assuming that the cycle lengths have to be the same.

$($b$)$ Determine the optimal rotation schedules assuming the cycle lengths do not have to be the same.

$($c$)$ Determine the optimal schedule if the schedule on machine $1$ has to be a rotation schedule and the schedule on machine $2$ may be an arbitrary schedule.

Please answer all parts with steps please :)