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In this study, we investigated several online and semi-online schedulingproblems on two hierarchical machines with a common due date to maximize thetotal early work. For the pure online case, we designed an optimal onlinealgorithm with a competitive ratio of $\sqrt 2$. For the case when the totalprocessing time is known, we proposed an optimal semi-online algorithm with acompetitive ratio of $\frac{4}{3}$. Additionally, for the cases when thelargest processing time is known, we gave optimal algorithms with a competitiveratio of $\frac{6}{5}$ if the largest job is a lower hierarchy one, and of$\sqrt 5-1$ if the largest job is a higher hierarchy one, respectively.

Online and semi-online scheduling on two hierarchical machines with a common due date to maximize the total early work