Abstract
We consider online hierarchical scheduling on m identical machines in shared manufacturing to minimize the total completion time. Each job has a unit-size processing time. The jobs arrive one by one and must be assigned to one of the m machines before the next job arrives. The jobs with a lower hierarchy can only be processed on the first k machines, and the jobs with a higher hierarchy can be processed on any one of the m machines. We first show that the lower bound of the problem is at least where Proposing a greedy algorithm, we show that its tight competitive ratio is by analyzing a set of instances that must contain a worst-case instance, which is different from the general method of calculating the competitive ratio. In addition, for the case where k = 1, we present an improved online algorithm with a tight competitive ratio of which is optimal for Numerical experiments show that the greedy online algorithm has good performance, especially when k approaches 1 or m.
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