The polynomial solvability of selected bicriteria scheduling problems on parallel machines with equal length jobs and release dates

We consider bicriteria scheduling problems involving several classical well-known scheduling objectives on parallel, identical machines with job release dates. The jobs are assumed to have equal processing times. Our bicriteria treatment of these problems includes lexicographic optimization, minimization of a composite linear function, generation of schedules on the efficient frontier and the generation of all Pareto optimal solutions. Using the framework provided by the dynamic program of Baptiste (2000), we show the polynomial solvability for several bicriteria problems when the number of machines is given. We also present algorithms that extend the linear programming approach of Brucker and Kravchenko (2008) and whose complexities are polynomial in the number of jobs and independent of the number of machines. The complexity status of the bicriteria problems considered in this paper was previously unknown.