Improving the Lagrangian relaxation approach for large job-shop scheduling

Lagrangian relaxation (LR) has recently emerged as a practical approach for job shop scheduling problems of realistic size. The efficiency of the approach, however, depends on how fast the dual problem is solved and how good the feasible solutions are constructed. The purpose of this paper is to address above two issues. First, a “variable target value” method is used to regulate the step size for surrogate subgradient optimization. The target values are updated iteratively whenever necessary, depending on the information obtained in the process of the algorithm. The convergence of the algorithm is proved using practically desirable step size rule. Then based on the insights of the old list scheduling algorithm, the SPT/CR priority index is selected in place of the incremental cost index. Testing results show that these modifications make the LR approach computationally more efficient and five to eight percent of duality gap improvement was obtained for large problems