Simple really
A check of the scheduled block times reveals:
ORD-MAN = 7hr 40 min (no 3rd pilot required, but close)
MAN-ORD = 8hr 25 min (3rd pilot required)
Since you need the 3rd pilot for leg back and you'd have to get him over there anyway, why not have him operate over as a relief pilot since it is a long sector anyway.