Online scheduling of unit length jobs on a batching machine to maximize the number of early jobs with lookahead

Abstract

This paper studies online scheduling of unit length jobs on a parallel batching machine with the help of lookahead. The objective is to maximize the number of early jobs. Denote by b the size of each batch with b = ∞ in the unbounded batching and b < ∞ in the bounded batching. In the L K L lookahead model, at a time instant t , an online algorithm can foresee all jobs that will arrive in the time segment ( t , t + L ) . When 0 ≤ L < 1 , we show that a simple greedy online algorithm (independent of the value of L ) has a best possible competitive ratio of 1 / min { n , b + 1 } , where n is the number of jobs. This means that lookahead is useless when 0 ≤ L < 1 . When 1 ≤ L < 2 , we establish the upper bounds 0.39 (for b = ∞ ) and 2 / 3 (for b < ∞ ) of competitive ratios, and provide an online algorithm of competitive ratios 1 / 4 (for b = ∞ ) and 1 / 5 (for b < ∞ ).