One company has two product types, Types I and II$.$ The precedence constraint is shown is the

following table, and the task times are shown in the format of Type I$/$Type II$.$ For example, for task

A$,$ the task time $6/7$ means $6$ time units for Type I product, and $7$ time units for Type II product.

Task IP $($Immediate Predecessors$)$ Task Times

Type I$/$Type II

A $--6/7$

B A $25/22$

C A $15/10$

D B$,$C $23/25$

E B $12/0$

F E $5/6$

G D $36/40$

H F$,$G $16/15$

I H $12/10$

For Type I product, the cycle $($or takt$)$ time is $50,$ and the best known solution uses four stations $($A$,$

B$,$ E$,$ F$),($C$,$ D$),($G$),$ and $($H$,$ I$).$ For Type II product, the cycle time is $40.$

$($a$)$ Balance the line assuming synchronous $($paced$)$ transfer. In this instance, all workstations must

be able to perform all the assigned tasks for either product within the cycle time.

$($b$)$ Suppose a line was to be built for just Type II product. Find an appropriate line balance with C $=$

$40.$ Compare the balance delay for the separate$-$line and mixed line solutions.

$($Hint: answer question $($b$)$ first, and then compare the solution to question $($b$)$ for Type II product

with the best known solution to Type I product.

This problem is to $($i$)$ directly check the concept of separate lines vs$.$ synchronized lines, and $($ii$)$

indirectly check the branch and bound concept in the tree structure of solution space.$)$