A Parallel Machine Scheduling Problem Maximizing Total Weighted Early Work

We consider the total weighted early work maximization problem on identical machines in parallel such that the weights are identical, or the due date is the same. First, we present an approach to solve the case with a fixed number of machines in pseudo-polynomial time. Then, we develop approximation algorithms for the two cases with identical weights and with a common due date. For the case with identical weights, furthermore, we show that the parallel-machine and a single-machine cases are strongly NP-hard and weakly NP-hard, respectively, even if the due date of each job is equal to the processing time multiplied by a constant.