Scheduling equal-length jobs on identical parallel machines

We study the situation where a set of n jobs with release dates and equal processing times have to be scheduled on m identical parallel machines. We show that, if the objective function can be expressed as the sum of n functions fi of the completion time Ci of each job Ji, the problem can be solved in polynomial time for any fixed value of m. The only restriction is that functions fi have to be non-decreasing and that for any pair of jobs (Ji, Jj), the function fi − fj has to be monotonous. This assumption holds for several standard scheduling objectives, such as the weighted sum of completion times or the total tardiness. Hence, the problems (Pm|pi = p, ri| ∑ wiCi) and (Pm|pi = p, ri| ∑ Ti) are polynomially solvable.