An alternative framework to Lagrangian relaxation approach for job shop scheduling

Lagrangian relaxation (LR) has emerged as a practical approach for complex scheduling problems. The efficiency of the approach, however, depends on how fast the relaxed subproblems and the dual problem are solved. Previously, machine capacity constraints were relaxed and the subproblems were solved by using dynamic programming (DP). The number of multipliers and the computation complexity of the DP algorithm, however, are proportional to the time horizon. This becomes a barrier for the approach to solve problems with long time horizons. The paper presents an alternative framework to overcome the barrier. By using much fewer multipliers to relax operation precedence constraints rather than machine capacity constraints and by approximately solving subproblems, a new LR approach is developed. The approach can find good schedules for problems with thousands of pairs and tens of machines within a short time on a personal computer, making it possible for practical use.