Fachbereich 3 Mathematik Polyhedral Approaches to Machine Scheduling Polyhedral Approaches to Machine Scheduling

We provide a review and synthesis of polyhedral approaches to nonpreemptive machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time{indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simpliied new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems.