Improved bounds for batch scheduling with nonidentical job sizes

Here, we revisit the bounded batch scheduling problem with nonidentical job sizes on single and parallel identical machines, with the objective of minimizing the makespan. For the single machine case, we present an algorithm which calls an online algorithm P (chosen arbitrarily) for the one-dimensional bin-packing problem as a sub-procedure, and prove that its worst-case ratio is the same as the absolute performance ratio of P. Hence, there exists an algorithm with worst-case ratio 17 10 , which is better than any known upper bound on this problem. For the parallel machines case, we prove that there does not exist any polynomial-time algorithm with worst-case ratio smaller than 2 unless P = NP, even if all jobs have unit processing time. Then we present an algorithm with worst-case ratio arbitrarily close to 2. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 351–358, 2014