Scheduling of manufacturing systems using the Lagrangian relaxation technique

The practical solutions for three manufacturing scheduling problems are examined. As each problem is formulated, constraints are added or modified to reflect increasing real world complexity. The first problem considers scheduling single-operation jobs on identical machines. The second problem is concerned with scheduling multiple-operation jobs with simple fork/join precedence constraints on identical machines. The third problem is the job shop problem in which multiple-operation jobs with general precedence constraints are scheduled on multiple machine types Langrangian relaxation is used to decompose each of the scheduling problems into job- or operation-level subproblems. The subproblems are easier to solve than the original problem and have intuitive appeal. This technique results in algorithms which generate near-optimal schedules efficiently, while giving a lower bound on the optimal cost. In resolving the scheduling problem from one time instant to the next, the Lagrange multipliers from the last schedule can be used to initialize the multipliers, further reducing the computation time