In this study, we investigate several online and semi-online scheduling problems related to two hierarchical machines with a common due date to maximize the total early work. For the pure online case, we design an optimal online algorithm with a competitive ratio of √2. Additionally, for the cases when the largest processing time is known, we give optimal algorithms with a competitive ratio of 6/5 if the largest job is a lower-hierarchy one, and of √5 − 1 if the largest job is a higher-hierarchy one.
We consider the scheduling problem on unrelated parallel machines in order to minimize the total late work. Since the problem is NP-hard, we propose a mathematical model and two dedicated exact approaches for solving it, based on the branching and bounding strategy and on enumerating combined with a dynamic programming algorithm. The time efficiencies of all three approaches are evaluated through computational experiments.
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