The two-machine flow shop problem with conflict graphs

Abstract

In this paper, we consider the problem of scheduling jobs on a two-machine flow shop subject to constraints represented by an undirected graph G, in which each edge joins a pair of conflicting jobs that cannot be processed simultaneously on different machines. The problem of minimizing the maximum completion time (makespan) is known to be NP-hard in the strong sense even when all the operations require one unit of processing time. We prove that the permutation schedules are not dominant even for two machines, and we present a special case for which an optimal schedule can be found by a permutation schedule. On the other hand, we propose four Mixed Integer Linear Programming (MILP) models alongside with an experimental study to measure their performance on a wide range of test problems.