Online early work scheduling on parallel machines

We consider non-preemptive online parallel-machine scheduling with a common due date to maximize the total early work of all the jobs, i.e., the total processing time of the jobs (or parts) completed before the common due date. For the general case of m machines, we provide a parameter lower bound with respect to m. For the online algorithm, we first show that the tight competitive ratio of the classical list scheduling (LS) algorithm is 43. We then improve the upper bound on the competitive ratio for the previous algorithm, EFFm, to 1.2956. Additionally, we present a formula to compute the upper bound on the competitive ratio for any given m. For the case of three machines, we improve the lower bound to 1.1878 and propose an improved online algorithm with a tight competitive ratio of 1.2483.