An optimization-based algorithm for job shop scheduling

Scheduling is a key factor for manufacturing productivity. Effective scheduling can improve on-time delivery, reduce inventory, cut lead times, and improve the utilization of bottleneck resources. Because of the combinatorial nature of scheduling problems, it is often difficult to find optimal schedules, especially within a limited amount of computation time. Production schedules therefore are usually generated by using heuristics in practice. However, it is very difficult to evaluate the quality of these schedules, and the consistency of performance may also be an issue. In this paper, near-optimal solution methodologies for job shop scheduling are examined. The problem is formulated as integer optimization with a ‘separable’ structure. The requirement of on-time delivery and low work-in-process inventory is modelled as a goal to minimize a weighted part tardiness and earliness penalty function. Lagrangian relaxation is used to decompose the problem into individual part subproblems with intuitive appeal. By iteratively solving these subproblems and updating the Lagrangian multipliers at the high level, near-optimal schedules are obtained with a lower bound provided as a byproduct. This paper reviews a few selected methods for solving subproblems and for updating multipliers. Based on the insights obtained, a new algorithm is presented that combines backward dynamic programming for solving low level subproblems and interleaved conjugate gradient method for solving the high level problem. The new method significantly improves algorithm convergence and solution quality. Numerical testing shows that the method is practical for job shop scheduling in industries.