Minimizing Maximum Promptness and Maximum Lateness on a Single Machine

We consider the following single-machine bicriteria scheduling problem. A set of n independent jobs has to be scheduled on a single machine that can handle only one job at a time and that is available from time zero onwards. Each job J_j requires processing during a given positive uninterrupted time p_j, and has a given target start time s_j and due date d_j with A ≤ d_j-s_j ≤ A + p_j for some constant A. For each job J_jj = 1,..., n, its promptness P_j is defined as the difference between the target start time and the actual start time, and its lateness L_j as the difference between the completion time and the due date. We consider the problem of finding a schedule that minimizes a function F of maximum promptness P_max = max_1≤j≤nP_j and maximum lateness L_max = max_1≤j≤nL_j; we assume that F is nondecreasing in both arguments. We present an On² algorithm for the variant in which idle time is not allowed and an On² log n algorithm for the special case in which the objective function is linear. We prove that the problem is NP-hard if neither of these restrictions is imposed. As a side result, we prove that the special case of minimizing maximum lateness subject to release dates that lie in the interval [d_j-p_j-A, d_j-A], for all j = 1,..., n and for some constant A, is solvable in On log n time if no machine idle time is allowed and in On² log n time if machine idle time is allowed.