Abstract

We consider semi-online scheduling of an unbounded parallel batch machine to minimize the makespan where, at the present time instant t , information on the first longest job arriving after t is known. In this paper online means that jobs arrive over time, J ∗ ( t ) denotes the first longest job arriving after t , and p ∗ ( t ) and r ∗ ( t ) denote the processing time and arrival time of J ∗ ( t ) , respectively. Given information p ∗ ( t ) , we present an online algorithm with a competitive ratio ( 5 − 5 ) / 2 ≈ 1.382 , and show that the algorithm is the best possible; furthermore, this algorithm generates at most two batches. This algorithm is also the best possible given information J ∗ ( t ) . Given information r ∗ ( t ) , we present an online algorithm with a competitive ratio 3 / 2 , and show that any online algorithm cannot have a competitive ratio less than 3 3 ≈ 1.442 ; furthermore, this algorithm generates at most three batches. Given information r ∗ ( t ) with the restriction that an online algorithm generates at most two batches, we present an online algorithm with a competitive ratio ( 5 + 1 ) / 2 ≈ 1.618 , and show that the algorithm is the best possible.

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