Parallel dedicated machines scheduling with chain precedence constraints

A set of n nonpreemptive tasks are to be scheduled on m parallel dedicated machines with a regular criterion. Chain precedence constraints among the tasks, deterministic processing times and processing machine of each task are given. We present computational complexity results and solution algorithms for some special cases. When the precedence relations among the tasks are given by two chains, we provide efficient solution algorithms for the minimization of the weighted sum of task completion times and the number of tardy jobs. Moreover, we show that when minimizing the sum of non-decreasing functions of the completion times of the tasks, a pseudopolynomial time algorithm and a fully polynomial time approximation scheme (FPTAS) exist. Indeed, when the objective is the minimization of the weighted number of tardy jobs or total weighted tardiness the problem is NP-hard in the ordinary sense. Finally, we consider slightly more general precedence structures and show that, even when the precedence constraints form a comb, makespan minimization problem turns out to be strongly NP-hard.