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