We study the special case of the m machine ??flow shop problem in which the pro??cessing time of each operation of job j is equal to pj; this variant of the fl??ow shop problem is known as the proportionate fl??ow shop problem We show that for any number of machines and for any regular performance criterion we can restrict our search for an optimal schedule to permutation schedules. Moreover,?? we show that the problem of minimizing total weighted completion time is solvable in O(n^2) time.
Key Words and Phrases: fl??ow shop scheduling,?? weighted completion time.