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.

$\backslash$table$[[$Items$,1,2,3,4,5],[D_j,60,60,80,80,100],[Q_j,200,200,300,300,400],[h_j,20,30,40,20,20],[c_j,3000,2000,800,4000,1500]]$

$($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.