Optimally Scheduling Small Numbers of Identical Parallel Machines

Given a set of n different jobs, each with an associated running time, and a set of k identical machines, our task is to assign each job to a machine to minimize the time to complete all jobs. In the OR literature, this is called identical parallel machine scheduling, while in AI it is called number partitioning. For eight or more machines, an OR approach based on bin packing appears best, while for fewer machines, a collection of AI search algorithms perform best. We focus here on scheduling up to seven machines, and make several new contributions. One is a new method that significantly reduces duplicate partitions for all values of k, including k = 2. Another is a new version of the Complete-Karmarkar-Karp (CKK) algorithm that minimizes the makespan. A surprising negative result is that dynamic programming is not competitive for this problem, even for k = 2. We also explore the effect of precision of values on the choice of the best algorithm. Despite the simplicity of this problem, a number of different algorithms have been proposed, and the most efficient algorithm depends on the number of jobs, the number of machines, and the precision of the running times.