The one-machine sequencing problem

Abstract

Let M 1 and M 3 be non-bottleneck machines and M 2 a bottleneck machine processing only one job at a time. Suppose that n jobs have to be processed on M 1 , M 2 and M 1 (in that order) and job i has to spend a time a , on M 1 , d 1 on M 2 and q 1 on M 3 : we want to minimize the makespan. This problem is important since its resolution provides a bound on the makespan of complicated systems such as job shops. It is NP-hard in the strong sense. However, efficient branch and bound methods exist and we describe one of them. Our bound for the tree-search is very close to the bound used by Florian et al., but the principle of branching is quite different. At every node, we construct by an O( n log n ) algorithm a Schrage schedule; then we define a critical job c , a critical set J and consider two subsets of schedules; the schedules when c procedes every job of J and the schedules when c follows every job of J . We give the results of this method and prove that the difference between the optimum and the Schrage schedule is less than d 1 .