A Lagrangian relaxation approach to job shop scheduling problems

An exploration is made of the use of Lagrangian relaxation to schedule job shops, which include multiple machine types, generic precedence constraint, and simple routing considerations. From an augmented Lagrangian formulation, a decomposed solution methodology is developed using a Jacobi-type iterative approach. The subgradient method and the multiplier method are used to update the multipliers and the penalty coefficients. The dual solution forms the basis of a list scheduling algorithm which generates a feasible schedule. Unfortunately, the dual cost is not a lower bound on the optimal cost because of the Jacobi iterative technique employed. In order to evaluate the schedule, a second problem formulation is adopted. Its solution would ordinarily require prohibitive memory and considerable computation time. By utilizing part of the multipliers obtained from the first problem formulation, however, an effective lower bound on the optimal cost can be quickly obtained. A numerical example is given in which schedule cost is within 2% of its lower bound