Parallel machine scheduling using Lagrangian relaxation

A two-level optimization methodology is presented for scheduling independent jobs with due dates on parallel machines (resources). A Lagrangian relaxation technique is applied to a constrained integer programming formulation of the problem. A decomposition by job of the dual problem serves to simplify the solution at a low level. The high-level problem is then solved by a subgradient method. In general, the resulting solution in the dual space is associated with an infeasible solution in the primal space. A heuristic approach is developed to generate a feasible solution, using the solution to the dual problem and concept of 'degree of freedom' in scheduling each job. On two problems tested, the feasible solutions generated by the heuristic are within 0.5% of the dual optima. The concept of degree of freedom can be used to study the scheduling of new jobs. The method can also be extended to general job shops.<>