A macro-level scheduling method using Lagrangian relaxation

In this paper, a macro-level scheduling method is developed to provide high-level planning support for factories with multiple coordinating cells. The key challenges are large problem sizes, complicated product process plans, stringent cell coordination requirements, and possible resource overload. To model the problem with manageable complexity, detailed operations of a product within a cell are aggregated as a single operation whose processing time is related to the amount of resources allocated. “Overload variables” are introduced and penalized in the objective function. The goal is to properly allocate resources, efficiently handle complicated process plans, and coordinate cells to ensure on-time delivery, low working-in-process inventory, and small resource overload. The formulation obtained is “separable” and can be effectively decomposed by using Lagrangian relaxation. A combined backward and forward dynamic programming (BFDP) method is developed to solve a product subproblem after a novel transformation of its process plan. The BFDP is further simplified and solved approximately following the idea of the “surrogate subgradient method” to reduce the computation requirements for large problems. Numerical results show that near-optimal schedules can be obtained for problems with up to 50 000 operations within a reasonable amount of computation time.