Online hierarchical parallel-machine scheduling in shared manufacturing to minimize the total completion time

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, 1 <= k <= m-1, 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 1+min{1/m-k+1,max{2/k inverted right perpendiculexpressionr s right ceiling +3k+2m+4/ inverted right perpendiculexpressionr s right ceiling ,2/k left floor s right floor +3k+2m+4/ left floor s right floor }}, where s=root 2m+4/k. Proposing a greedy algorithm, we show that its tight competitive ratio is 1+2(m-k)k/m(root(4m-3k)k root+k)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 1+max{2/ inverted right perpendiculexpressionr root 2m+4 root right ceiling +2m+4/ inverted right perpendiculexpressionr root 2m+4 root right ceiling +3,2/ left floor root 2m+4 right floor +2m+4/ left floor root 2m+4 root right floor +3}, which is optimal for 2 <= m <= 5. Numerical experiments show that the greedy online algorithm has good performance, especially when k approaches 1 or m.