Scheduling equal-length jobs on identical parallel machines

Abstract

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 f_i of the completion time C_i of each job J_i, the problem can be solved in polynomial time for any fixed value of m. The only restriction is that functions f_i have to be non-decreasing and that for any pair of jobs (J_i,J_j), the function f_i−f_j 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|p_i=p,r_i|∑w_iC_i) and (Pm|p_i=p,r_i|∑T_i) are polynomially solvable.