Minimizing the number of late multi-task jobs on identical machines in parallel

We consider the problem of scheduling multi-task jobs on identical machines in parallel. Each multi-task job consists of one or more tasks. Each job has a release date and a due date. A task of a job can be processed by any one of the machines. Multiple machines can process the tasks of a job concurrently. The completion time of a job is the time at which all its individual tasks have been completed. A job is late if it is completed after its due date. We study the problem of minimizing the total number of late jobs. We show that while some special cases are solvable, the general problem is NP-hard and there exists no polynomial time ρ-approximation algorithm, for any ρ > 1. We present a general algorithm for the problem and derive from it six heuristics whose performance is evaluated by experimental results.