Multiprocessor task scheduling to minimize the maximum tardiness and the total completion time

We consider a scheduling problem with two parallel processors, where one task may need the processing of more than one processor simultaneously. Alternatives are available to allocate processors to process a task. Under a given alternative, a task is assigned to some specific processor(s), and requires correspondingly a specific processing time. Task preemption is allowed. The problem is to minimize the maximum tardiness of task completion times from their due dates, through determining the following: 1) the optimal assignment of each task to processor(s) and 2) the optimal schedule for each processor to process the tasks assigned to it. We present a two-stage approach, which can generate an optimal solution in pseudopolynomial time. We also apply/extend the two-stage approach to other problems where the number of processors required by each task is prespecified, preemption is prohibited, and the total completion time is to be minimized.